= Solution
Neglecting the rapidly oscillating radiation source on subhorizon scales, <stress-energy conservation> and the trace of the <Einstein field equations> give
$$
\delta_m''+\mathcal H\delta_m'-4\pi Ga^2\rho_m\delta_m=0.
$$
Equivalently in <cosmic time>, $\delta_{m,tt}+2H\delta_{m,t}-4\pi G\rho_m\delta_m=0$. Radiation still contributes to the background expansion, even when its perturbation source is averaged away.
Put $\eta=a/a_{\rm eq}$ and let $\rho_{\rm eq}$ be the density of either component at <matter-radiation equality>. Then $\rho_m=\rho_{\rm eq}\eta^{-3}$ and $\rho_r=\rho_{\rm eq}\eta^{-4}$. With $C=(8\pi G/3)\rho_{\rm eq}a_{\rm eq}^2$, the <Friedmann equation> gives
$$
\mathcal H^2=C\frac{1+\eta}{\eta^2},\quad
(\eta')^2=C(1+\eta),\quad \eta''=\frac C2,\quad
4\pi Ga^2\rho_m=\frac{3C}{2\eta}.
$$
Apply the <chain rule> to the density equation:
$$
C(1+\eta)\delta_{\eta\eta}+C\left(\frac12+\frac{1+\eta}{\eta}\right)\delta_\eta-\frac{3C}{2\eta}\delta=0.
$$
Thus the <Mészáros equation> is
$$
\boxed{2\eta(1+\eta)\delta_{\eta\eta}+(2+3\eta)\delta_\eta-3\delta=0.}
$$
Substituting $D_+=1+3\eta/2$ gives $(2+3\eta)(3/2)-3(1+3\eta/2)=0$, so
$$
\boxed{\delta_m=A(\mathbf x)(1+3\eta/2)}
$$
is indeed a solution. It is approximately constant during <radiation domination> and grows as $a$ during <matter domination>. The second independent solution is
$$
D_-=(1+3\eta/2)\log\frac{\sqrt{1+\eta}+1}{\sqrt{1+\eta}-1}-3\sqrt{1+\eta},
$$
which behaves as $\log(4/\eta)-3$ at small $\eta$. Thus smoothing radiation does not eliminate every logarithmic solution; matching through actual horizon entry fixes the combination excited.
Modes with $k\ll k_{\rm eq}$ enter after equality and retain the $P(k)\propto k$ shape. Modes with $k\gg k_{\rm eq}$ enter during radiation domination: their horizon-entry spectrum is $P_H\propto k^{-3}$ and they gain little growth until equality. The late <matter power spectrum> therefore turns over near $k_{\rm eq}$ and has \b[a large-scale $k$ branch and a small-scale $k^{-3}$ branch], with the more accurate radiation-era matching giving
$$
P(k)\propto k^{-3}\log^2(k/k_{\rm eq})\quad(k\gg k_{\rm eq}).
$$
Equivalently the <cold-dark-matter transfer function> falls approximately as $(k_{\rm eq}/k)^2$, up to the logarithm. The turnover is the consequence of the <Mészáros effect>; it is not a change of the primordial scale-invariance condition.
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